with \kappa\rightarrow\kappa(\kappa_0,\kappa_1,\kappa_2) and
where \Omega denotes the whole domain, \partial\Omega denotes
the boundary of the domain such that these parameters verify the observation
u_\text{obs} .

In other words, the goal is to minimize an objective function J(\kappa_0,\kappa_1,\kappa_2)

In practice, observations are often boundary measurement taken on a subset of the
domain boundary U\subseteq\partial\Omega . Thus arise concerns about existence
and uniqueness of solution for the inverse problem, especially if observations
does not cover the whole boundary.

we expand the perturbed diffusion coefficient as \kappa^\delta=\kappa +\alpha\delta\kappa
and rearrange terms by u^\delta -u ( =\alpha\delta u ).
We divide by \alpha as previously and makes \alpha\rightarrow 0 , we
have